If, for some \(m \ge N_{\delta }\) , \(H(t,\cdot ,p^{m}(t),q^{m}(t),\cdot )\)  is convex for a.e. \(t\in [0,T]\) , \(\mathbb {P}\) -a.s., then it is equivalent to the concavity of \(-H(t,\cdot ,p^{m}(t),q^{m}(t),\cdot )\)  for a.e. \(t\in [0,T]\) , \(\mathbb {P}\) -a.s.. Under Assumption REF , this and (REF ) satisfy the conditions in Theorem 5.1 in [1]}. Consequently, there exists a constant \(\tilde{C}>0\)  independent of \(\delta \)  and \(m\)  such that
\(J(u^{m}(\cdot ))-\inf _{u(\cdot ) \in \mathcal {U}[0,T]}J(u(\cdot ))\le \tilde{C}\delta ^{\frac{1}{2}},\) 
